Solving PDE with method of characteristcs with both initial conditions $0$

Question

I need to solve using the method of characteristics: $$\frac{\partial ^2 u}{\partial t^2}=c^2\frac{\partial^2 u}{\partial x^2}$$ $$u(x,0)=0, \hspace{2cm} u(0,t)=h(t)$$ $$\frac{\partial u}{\partial t}(x,0) = 0, \hspace{2cm} u(L,t) = 0$$ I'm at a loss how to do this. Normally, you would extend the inital conditions as odd or even periodic functions and proceed from there. But now both the initial conditions are $0$. If you say that $u(x,t) = F(x-ct)+G(x+ct)$, then you'd find only zero as a solution. I get that this is not correct since $h(t)$, the boundary condition at $0$, is constantly "making" waves which are being reflected at the fixed end $L$. So how should I solve this then?